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Simplifying z2 + -5z = 30 Reorder the terms: -5z + z2 = 30 Solving -5z + z2 = 30 Solving for variable 'z'. Reorder the terms: -30 + -5z + z2 = 30 + -30 Combine like terms: 30 + -30 = 0 -30 + -5z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '30' to each side of the equation. -30 + -5z + 30 + z2 = 0 + 30 Reorder the terms: -30 + 30 + -5z + z2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + -5z + z2 = 0 + 30 -5z + z2 = 0 + 30 Combine like terms: 0 + 30 = 30 -5z + z2 = 30 The z term is -5z. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5z + 6.25 + z2 = 30 + 6.25 Reorder the terms: 6.25 + -5z + z2 = 30 + 6.25 Combine like terms: 30 + 6.25 = 36.25 6.25 + -5z + z2 = 36.25 Factor a perfect square on the left side: (z + -2.5)(z + -2.5) = 36.25 Calculate the square root of the right side: 6.020797289 Break this problem into two subproblems by setting (z + -2.5) equal to 6.020797289 and -6.020797289.Subproblem 1
z + -2.5 = 6.020797289 Simplifying z + -2.5 = 6.020797289 Reorder the terms: -2.5 + z = 6.020797289 Solving -2.5 + z = 6.020797289 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = 6.020797289 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = 6.020797289 + 2.5 z = 6.020797289 + 2.5 Combine like terms: 6.020797289 + 2.5 = 8.520797289 z = 8.520797289 Simplifying z = 8.520797289Subproblem 2
z + -2.5 = -6.020797289 Simplifying z + -2.5 = -6.020797289 Reorder the terms: -2.5 + z = -6.020797289 Solving -2.5 + z = -6.020797289 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = -6.020797289 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = -6.020797289 + 2.5 z = -6.020797289 + 2.5 Combine like terms: -6.020797289 + 2.5 = -3.520797289 z = -3.520797289 Simplifying z = -3.520797289Solution
The solution to the problem is based on the solutions from the subproblems. z = {8.520797289, -3.520797289}
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